Theorem gsumhashmul 33046

Description: Express a group sum by grouping by nonzero values. (Contributed by Thierry Arnoux, 22-Jun-2024.)

Hypotheses

Ref	Expression
gsumhashmul.b	⊢ 𝐵 = (Base‘𝐺)
gsumhashmul.z	⊢ 0 = (0g‘𝐺)
gsumhashmul.x	⊢ · = (.g‘𝐺)
gsumhashmul.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumhashmul.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumhashmul.1	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsumhashmul	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))

Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥

Allowed substitution hint:   · (𝑥)

Proof of Theorem gsumhashmul

Dummy variables 𝑡 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumhashmul.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		suppssdm 8200	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
3	2, 1	fssdm 6755	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
4	1, 3	feqresmpt 6977	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥)))
5	4	oveq2d 7446	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))))
6		gsumhashmul.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
7		gsumhashmul.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
8		gsumhashmul.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
9		gsumhashmul.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp 0 )
10		relfsupp 9400	. . . . . . . . 9 ⊢ Rel finSupp
11	10	brrelex1i 5744	. . . . . . . 8 ⊢ (𝐹 finSupp 0 → 𝐹 ∈ V)
12	9, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
13	1	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14	12, 13	fndmexd 7926	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
15		ssidd 4018	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
16	6, 7, 8, 14, 1, 15, 9	gsumres 19945	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹))
17		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(1st ‘𝑧))
18		fveq2 6906	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑧) → (𝐹‘𝑥) = (𝐹‘(1st ‘𝑧)))
19	9	fsuppimpd 9406	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
20		ssidd 4018	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
21	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴⟶𝐵)
22	3	sselda 3994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ 𝐴)
23	21, 22	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹‘𝑥) ∈ 𝐵)
24	1	ffund 6740	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
25		funrel 6584	. . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹)
26		reldif 5827	. . . . . . . . 9 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ (V × { 0 })))
27	24, 25, 26	3syl 18	. . . . . . . 8 ⊢ (𝜑 → Rel (𝐹 ∖ (V × { 0 })))
28		1stdm 8063	. . . . . . . 8 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
29	27, 28	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ dom (𝐹 ∖ (V × { 0 })))
30	7	fvexi 6920	. . . . . . . . . . . 12 ⊢ 0 ∈ V
31	30	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ V)
32		fressupp 32702	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
33	24, 12, 31, 32	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
34	33	dmeqd 5918	. . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0 })))
35	2	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹)
36		ssdmres 6032	. . . . . . . . . 10 ⊢ ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
37	35, 36	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
38	34, 37	eqtr3d 2776	. . . . . . . 8 ⊢ (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
39	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 ))
40	29, 39	eleqtrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (1st ‘𝑧) ∈ (𝐹 supp 0 ))
41	24	funresd 6610	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 )))
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
43	37	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 )))
44	43	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )))
45		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 ))
46	45	fvresd 6926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥))
47		funopfvb 6962	. . . . . . . . . . 11 ⊢ ((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥) ↔ ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 ))))
48	47	biimpa 476	. . . . . . . . . 10 ⊢ (((Fun (𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥)) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
49	42, 44, 46, 48	syl21anc 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ↾ (𝐹 supp 0 )))
50	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 })))
51	49, 50	eleqtrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ (𝐹 ∖ (V × { 0 })))
52		eqeq2 2746	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → (𝑧 = 𝑣 ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
53	52	bibi2d 342	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → ((𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
54	53	ralbidv 3175	. . . . . . . . 9 ⊢ (𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩ → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	54	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = ⟨𝑥, (𝐹‘𝑥)⟩) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)))
56		fvexd 6921	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ V)
57	27	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → Rel (𝐹 ∖ (V × { 0 })))
58		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
59		1st2nd 8062	. . . . . . . . . . . . . . . . 17 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
60	57, 58, 59	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
61		opeq1 4877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ⟨𝑥, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
62	61	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ⟨𝑥, (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
63	60, 62	eqtr4d 2777	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩)
64		difssd 4146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆ 𝐹)
65	64	sselda 3994	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹)
66	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ 𝐹)
67	63, 66	eqeltrrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹)
68	63, 67	jca 511	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩ ∧ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹))
69		opeq2 4878	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, (2nd ‘𝑧)⟩)
70	69	eqeq2d 2745	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩))
71	69	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹))
72	70, 71	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (𝑧 = ⟨𝑥, (2nd ‘𝑧)⟩ ∧ ⟨𝑥, (2nd ‘𝑧)⟩ ∈ 𝐹)))
73	56, 68, 72	spcedv 3597	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
74		vex 3481	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
75	74	elsnres 6040	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
76	73, 75	sylibr 234	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥}))
77	13	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝐹 Fn 𝐴)
78	22	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑥 ∈ 𝐴)
79		fnressn 7177	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
80	77, 78, 79	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝐹 ↾ {𝑥}) = {⟨𝑥, (𝐹‘𝑥)⟩})
81	76, 80	eleqtrd 2840	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ {⟨𝑥, (𝐹‘𝑥)⟩})
82		elsni 4647	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {⟨𝑥, (𝐹‘𝑥)⟩} → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
83	81, 82	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
84		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩)
85	84	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → (1st ‘𝑧) = (1st ‘⟨𝑥, (𝐹‘𝑥)⟩))
86		fvex 6919	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
87	74, 86	op1st 8020	. . . . . . . . . . 11 ⊢ (1st ‘⟨𝑥, (𝐹‘𝑥)⟩) = 𝑥
88	85, 87	eqtr2di 2791	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩) → 𝑥 = (1st ‘𝑧))
89	83, 88	impbida 801	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
90	89	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = ⟨𝑥, (𝐹‘𝑥)⟩))
91	51, 55, 90	rspcedvd 3623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣))
92		reu6 3734	. . . . . . 7 ⊢ (∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st ‘𝑧) ↔ ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣))
93	91, 92	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st ‘𝑧))
94	17, 6, 7, 18, 8, 19, 20, 23, 40, 93	gsummptf1o 19995	. . . . 5 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))))
95	5, 16, 94	3eqtr3d 2782	. . . 4 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))))
96		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
97	96	eldifad 3974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹)
98		funfv1st2nd 8069	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝐹‘(1st ‘𝑧)) = (2nd ‘𝑧))
99	24, 97, 98	syl2an2r 685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st ‘𝑧)) = (2nd ‘𝑧))
100	99	mpteq2dva 5247	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧)))
101	100	oveq2d 7446	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st ‘𝑧)))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))))
102	95, 101	eqtrd 2774	. . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))))
103		nfcv 2902	. . . 4 ⊢ Ⅎ𝑧(1st ‘𝑡)
104		fvex 6919	. . . . 5 ⊢ (2nd ‘𝑡) ∈ V
105		fvex 6919	. . . . 5 ⊢ (1st ‘𝑡) ∈ V
106	104, 105	op2ndd 8023	. . . 4 ⊢ (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ → (2nd ‘𝑧) = (1st ‘𝑡))
107		resfnfinfin 9374	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
108	13, 19, 107	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
109	33, 108	eqeltrrd 2839	. . . 4 ⊢ (𝜑 → (𝐹 ∖ (V × { 0 })) ∈ Fin)
110	33	rneqd 5951	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0 })))
111		rnresss 6036	. . . . . 6 ⊢ ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹
112	1	frnd 6744	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
113	111, 112	sstrid 4006	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
114	110, 113	eqsstrrd 4034	. . . 4 ⊢ (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵)
115		2ndrn 8064	. . . . 5 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd ‘𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
116	27, 115	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (2nd ‘𝑧) ∈ ran (𝐹 ∖ (V × { 0 })))
117		relcnv 6124	. . . . . . . 8 ⊢ Rel ◡𝐹
118		reldif 5827	. . . . . . . 8 ⊢ (Rel ◡𝐹 → Rel (◡𝐹 ∖ ({ 0 } × V)))
119	117, 118	mp1i 13	. . . . . . 7 ⊢ (𝜑 → Rel (◡𝐹 ∖ ({ 0 } × V)))
120		1st2nd 8062	. . . . . . 7 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
121	119, 120	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 = ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩)
122		cnvdif 6165	. . . . . . . . . 10 ⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ◡(V × { 0 }))
123		cnvxp 6178	. . . . . . . . . . 11 ⊢ ◡(V × { 0 }) = ({ 0 } × V)
124	123	difeq2i 4132	. . . . . . . . . 10 ⊢ (◡𝐹 ∖ ◡(V × { 0 })) = (◡𝐹 ∖ ({ 0 } × V))
125	122, 124	eqtri 2762	. . . . . . . . 9 ⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ({ 0 } × V))
126	125	eqimss2i 4056	. . . . . . . 8 ⊢ (◡𝐹 ∖ ({ 0 } × V)) ⊆ ◡(𝐹 ∖ (V × { 0 }))
127	126	a1i 11	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆ ◡(𝐹 ∖ (V × { 0 })))
128	127	sselda 3994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → 𝑡 ∈ ◡(𝐹 ∖ (V × { 0 })))
129	121, 128	eqeltrrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ ◡(𝐹 ∖ (V × { 0 })))
130	105, 104	opelcnv 5894	. . . . 5 ⊢ (⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ ◡(𝐹 ∖ (V × { 0 })) ↔ ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
131	129, 130	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ∈ (𝐹 ∖ (V × { 0 })))
132	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0 })))
133		eqidd 2735	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} = ∪ ◡{𝑧})
134		cnvf1olem 8133	. . . . . . . . 9 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ ∪ ◡{𝑧} = ∪ ◡{𝑧})) → (∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪ ◡{∪ ◡{𝑧}}))
135	134	simpld 494	. . . . . . . 8 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ ∪ ◡{𝑧} = ∪ ◡{𝑧})) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })))
136	132, 96, 133, 135	syl12anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })))
137	136, 125	eleqtrdi 2848	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ (◡𝐹 ∖ ({ 0 } × V)))
138		eqeq2 2746	. . . . . . . . 9 ⊢ (𝑢 = ∪ ◡{𝑧} → (𝑡 = 𝑢 ↔ 𝑡 = ∪ ◡{𝑧}))
139	138	bibi2d 342	. . . . . . . 8 ⊢ (𝑢 = ∪ ◡{𝑧} → ((𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
140	139	ralbidv 3175	. . . . . . 7 ⊢ (𝑢 = ∪ ◡{𝑧} → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
141	140	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = ∪ ◡{𝑧}) → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧})))
142	117, 118	mp1i 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → Rel (◡𝐹 ∖ ({ 0 } × V)))
143		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)))
144		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
145		df-rel 5695	. . . . . . . . . . . . . 14 ⊢ (Rel (◡𝐹 ∖ ({ 0 } × V)) ↔ (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
146	119, 145	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
147	146	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
148	147, 143	sseldd 3995	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 ∈ (V × V))
149		2nd1st 8061	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (V × V) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
150	148, 149	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
151	144, 150	eqtr4d 2777	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑧 = ∪ ◡{𝑡})
152		cnvf1olem 8133	. . . . . . . . . 10 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪ ◡{𝑡})) → (𝑧 ∈ ◡(◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = ∪ ◡{𝑧}))
153	152	simprd 495	. . . . . . . . 9 ⊢ ((Rel (◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪ ◡{𝑡})) → 𝑡 = ∪ ◡{𝑧})
154	142, 143, 151, 153	syl12anc 837	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩) → 𝑡 = ∪ ◡{𝑧})
155	27	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → Rel (𝐹 ∖ (V × { 0 })))
156	96	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 })))
157		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 = ∪ ◡{𝑧})
158		cnvf1olem 8133	. . . . . . . . . . 11 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = ∪ ◡{𝑧})) → (𝑡 ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪ ◡{𝑡}))
159	158	simprd 495	. . . . . . . . . 10 ⊢ ((Rel (𝐹 ∖ (V × { 0 })) ∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧ 𝑡 = ∪ ◡{𝑧})) → 𝑧 = ∪ ◡{𝑡})
160	155, 156, 157, 159	syl12anc 837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 = ∪ ◡{𝑡})
161	146	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V × V))
162		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)))
163	161, 162	sseldd 3995	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑡 ∈ (V × V))
164	163, 149	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → ∪ ◡{𝑡} = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
165	160, 164	eqtrd 2774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪ ◡{𝑧}) → 𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
166	154, 165	impbida 801	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) → (𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧}))
167	166	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = ∪ ◡{𝑧}))
168	137, 141, 167	rspcedvd 3623	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢))
169		reu6 3734	. . . . 5 ⊢ (∃!𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ ∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩ ↔ 𝑡 = 𝑢))
170	168, 169	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))𝑧 = ⟨(2nd ‘𝑡), (1st ‘𝑡)⟩)
171	103, 6, 7, 106, 8, 109, 114, 116, 131, 170	gsummptf1o 19995	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (2nd ‘𝑧))) = (𝐺 Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡))))
172		fveq2 6906	. . . . . 6 ⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧))
173	172	cbvmptv 5260	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡)) = (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑧))
174	33	cnveqd 5888	. . . . . . 7 ⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) = ◡(𝐹 ∖ (V × { 0 })))
175	174, 125	eqtr2di 2791	. . . . . 6 ⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) = ◡(𝐹 ↾ (𝐹 supp 0 )))
176	175	mpteq1d 5242	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑧)) = (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧)))
177	173, 176	eqtrid 2786	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡)) = (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧)))
178	177	oveq2d 7446	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦ (1st ‘𝑡))) = (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))))
179	102, 171, 178	3eqtrd 2778	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))))
180		nfcv 2902	. . 3 ⊢ Ⅎ𝑦(1st ‘𝑧)
181		nfv 1911	. . 3 ⊢ Ⅎ𝑥𝜑
182		vex 3481	. . . 4 ⊢ 𝑦 ∈ V
183	74, 182	op1std 8022	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
184		relcnv 6124	. . . 4 ⊢ Rel ◡(𝐹 ↾ (𝐹 supp 0 ))
185	184	a1i 11	. . 3 ⊢ (𝜑 → Rel ◡(𝐹 ↾ (𝐹 supp 0 )))
186		cnvfi 9214	. . . 4 ⊢ ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
187	108, 186	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
188	112	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹 ⊆ 𝐵)
189	184	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → Rel ◡(𝐹 ↾ (𝐹 supp 0 )))
190		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )))
191		1stdm 8063	. . . . . . 7 ⊢ ((Rel ◡(𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )))
192	189, 190, 191	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )))
193		df-rn 5699	. . . . . 6 ⊢ ran (𝐹 ↾ (𝐹 supp 0 )) = dom ◡(𝐹 ↾ (𝐹 supp 0 ))
194	192, 193	eleqtrrdi 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 )))
195	111, 194	sselid 3992	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ ran 𝐹)
196	188, 195	sseldd 3995	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st ‘𝑧) ∈ 𝐵)
197	180, 181, 6, 183, 185, 187, 8, 196	gsummpt2d 33034	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st ‘𝑧))) = (𝐺 Σg (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))))
198		df-ima 5701	. . . . . . 7 ⊢ (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 ))
199		supppreima 32705	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
200	24, 12, 31, 199	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
201	200	imaeq2d 6079	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))))
202	198, 201	eqtr3id 2788	. . . . . 6 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))))
203		funimacnv 6648	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
204	24, 203	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹))
205		difssd 4146	. . . . . . 7 ⊢ (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹)
206		dfss2 3980	. . . . . . 7 ⊢ ((ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
207	205, 206	sylib 218	. . . . . 6 ⊢ (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 }))
208	202, 204, 207	3eqtrd 2778	. . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
209	193, 208	eqtr3id 2788	. . . 4 ⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
210	8	cmnmndd 19836	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
211	210	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
212	108	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
213		imafi2 32728	. . . . . . 7 ⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
214	212, 186, 213	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin)
215	193, 113	eqsstrrid 4044	. . . . . . 7 ⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵)
216	215	sselda 3994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ 𝐵)
217		gsumhashmul.x	. . . . . . 7 ⊢ · = (.g‘𝐺)
218	6, 217	gsumconst 19966	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥 ∈ 𝐵) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
219	211, 214, 216, 218	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
220		cnvresima 6251	. . . . . . . 8 ⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 ))
221	209	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 })))
222	221	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 }))
223	222	snssd 4813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 }))
224		sspreima 7087	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
225	24, 223, 224	syl2an2r 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
226	200	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 })))
227	225, 226	sseqtrrd 4036	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ))
228		dfss2 3980	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥}))
229	227, 228	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥}))
230	220, 229	eqtr2id 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) = (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))
231	230	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (♯‘(◡𝐹 “ {𝑥})) = (♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})))
232	231	oveq1d 7445	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → ((♯‘(◡𝐹 “ {𝑥})) · 𝑥) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥))
233	219, 232	eqtr4d 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))
234	209, 233	mpteq12dva 5236	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥)))
235	234	oveq2d 7446	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))
236	179, 197, 235	3eqtrd 2778	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((♯‘(◡𝐹 “ {𝑥})) · 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  ∃!wreu 3375  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  {csn 4630  ⟨cop 4636  ∪ cuni 4911   class class class wbr 5147   ↦ cmpt 5230   × cxp 5686  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   ↾ cres 5690   “ cima 5691  Rel wrel 5693  Fun wfun 6556   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011   supp csupp 8183  Fincfn 8983   finSupp cfsupp 9398  ♯chash 14365  Basecbs 17244  0_gc0g 17485   Σ_g cgsu 17486  Mndcmnd 18759  ._gcmg 19097  CMndccmn 19812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814

This theorem is referenced by:  elrspunidl  33435